On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians

Abstract

We answer in the affirmative a question posed by V. Maz'ya of whether one can continue as a meromorphic function of $t$ the series representation of the fundamental solution of a certain nonlocal parabolic equation associated to a logarithmic Laplacian on the circle, which arises in the study of boundary value problems associated to the ordinary Laplacian on domains with thin cavities. The $a\ln(n) +O(1)$ growth of the eigenvalues of the integral operator, together with explicit formulas for the eigenfunctions and the subleading asymptotic behavior of the eigenvalues, allows us to show that the fundamental solution is reminiscent of a sum of shifted Riemann zeta functions or polylogarithms, depending on the spatial variable. We show an analogous result for an operator related to a different logarithmic Laplacian on the interval, whose structure is similar. Along the way we are led to prove and to conjecture a number of curious identities involving Bell polynomials and Bernoulli numbers related to the exponential of the digamma function which are of independent interest.